In a book I read this about existence of the solutions to parabolic PDEs:

the approximate solution $u_n(t)$ solves the equation
$$(u_n', w_j) + (Au_n, w_j) = \langle f, w_j \rangle\tag{1}$$
for $j=1,...,N$. Here $w_j$ is the set of basis functions associated with the problem. We can write this as
$$u_n' + Au_n = f$$
as an equality in $L^2(0,T;V').$

Surely this is not an equality in $L^2(0,T;V')$ but an equality in (I think) $L^2(0,T;V_n')$ where $V_n$ is the finite dimensional subset of $V$ spanned on $w_j$ for $j=1,...,N$? Because we only have that $(1)$ holds for everything in $V_n$ (by linearity).

The author then obtains a bound on $u_n'$ using this equality -- any elaboration would be useful.

1 Answer
1

First of all: I'm not sure what you mean by the expression $(Au,u)$. I'm only familiar with the notation $(A\nabla u, \nabla u)$ for elliptic operators. Or do you want to have also lower order terms and replace this expression by a bilinear form $B(u,u,t)$?